Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6,9,10)(7,11,12,8)(13,14)(15,19)(16,24)(17,22)(18,20)(21,26)(23,28)(25,27);(1,2)(3,4)(6,10)(7,12)(13,15,18,17)(14,19,20,22)(16,23,26,25)(21,27,24,28);(5,7)(6,11)(8,10)(9,12)(14,20)(19,22)(21,24)(27,28);(5,8)(6,12)(7,10)(9,11)(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,26)(25,28);(5,8)(6,7)(9,11)(10,12)(13,16)(14,21)(15,23)(17,25)(18,26)(19,27)(20,24)(22,28);(5,7)(6,11)(8,10)(9,12)(13,17,18,15)(14,22,20,19)(16,25,26,23)(21,28,24,27)Polycyclic presentation:
2^9|a^2=i,b^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72238149, #72238169, #72238209, … |
| maximal normal subgroup of | #72238149, #72238169, #72238209, … |
| maximal quotient of | #72297193, #72297574, #72297611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7875795);